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Angle Bracket — Definition, Formula & Examples

Angle brackets are the symbols ⟨ ⟩ used in mathematics to enclose specific objects such as ordered pairs, inner products, or averages. They look similar to the less-than and greater-than signs < > but serve a different purpose.

The angle bracket notation ⟨ ⟩\langle \, \rangle denotes a paired delimiter used to represent constructs including the inner product of two vectors ⟨u,v⟩\langle \mathbf{u}, \mathbf{v} \rangle, an ordered tuple ⟨a1,a2,…,an⟩\langle a_1, a_2, \ldots, a_n \rangle, or the expected value / average of a quantity ⟨x⟩\langle x \rangle in certain scientific and mathematical contexts.

How It Works

Angle brackets always come in a matched pair: a left bracket ⟨\langle and a right bracket ⟩\rangle. The meaning depends on context. In linear algebra, ⟨u,v⟩\langle \mathbf{u}, \mathbf{v} \rangle typically means the inner (dot) product of two vectors. In physics and statistics, ⟨x⟩\langle x \rangle often represents the mean or expected value of xx. Some authors also use angle brackets instead of parentheses to write ordered sequences or tuples.

Worked Example

Problem: Find the inner product ⟨u,v⟩\langle \mathbf{u}, \mathbf{v} \rangle where u=(2,3)\mathbf{u} = (2, 3) and v=(4,1)\mathbf{v} = (4, 1).
Step 1: The inner product (dot product) multiplies corresponding components and sums the results.
⟨u,v⟩=u1v1+u2v2\langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 + u_2 v_2
Step 2: Substitute the values and compute.
⟨u,v⟩=(2)(4)+(3)(1)=8+3=11\langle \mathbf{u}, \mathbf{v} \rangle = (2)(4) + (3)(1) = 8 + 3 = 11
Answer: ⟨u,v⟩=11\langle \mathbf{u}, \mathbf{v} \rangle = 11

Why It Matters

Angle bracket notation appears throughout linear algebra, quantum mechanics, and advanced statistics. Recognizing these symbols early prevents confusion when you encounter inner products in precalculus or physics courses.

Common Mistakes

Mistake: Using the less-than and greater-than signs < > in place of true angle brackets ⟨ ⟩.
Correction: The symbols < and > are inequality operators, not delimiters. In handwriting the shapes overlap, but in typed work use the proper angle bracket characters or the LaTeX commands \langle and \rangle to avoid ambiguity.